Understanding abstract algebra concepts

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Understanding abstract algebra concepts

Anna S. Titova

University of New Hampshire, Durham

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Titova, Anna S., "Understanding abstract algebra concepts" (2007). Doctoral Dissertations. 362.

UNDERSTANDING ABSTRACT ALGEBRA CONCEPTS

BY

ANNA S. TITOVA

BS, Russian State Pedagogical University, St. Petersburg, Russia, 1998 MS, University of New Hampshire, 2005

DISSERTATION

Submitted to the University of New Hampshire In Partial Fulfillment of

The Requirements for the Degree of

Doctor of Philosophy In

Mathematics Education

UMI Number: 3260585

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Dissertation Director, Dr. Sonia Hristovitch Assistant Professor of Mathematics

vs Dr. Karen J. Graham Professor of Mathematics

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Associate Professor of Mathematics

Dr. Edward Hinson

Associate Professor of Mathematics

Dr. Maria Basterra

Assistant Professor of Mathematics

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DEDICATION

for Dasha and Andrei Smuk; Ludmila Titova, Sergei Titov; Nina Nenasheva, Ivan Nenashev

who make me and my life possible,

ACKNOWLEDGEMENTS

Following the thesis manual, I only write my name on the front page of this dissertation. However, it would not be possible without help, support and guidance of many people. Writing the dissertation was a great challenge and now I owe my gratitude to all of those who intentionally or unintentionally assisted me during my graduate research and those who inflamed me for this study.

I would like to express my sincere gratitude to Professor Sonia Hristovitch who was advising my study and me during all these years. She was able to teach me what research is, how to conduct it, and what it means to write a dissertation. Her knowledge and positive attitude motivated and inspired me.

Special thanks go to Professor Karen Graham for her support during my doctoral study, especially during the most difficult first years. I am grateful to her for giving me the feedbacks and helping me to organize the thesis.

I would like to thank Professor Edward Hinson for all his help with the data collection for this study. It would not be possible for me to have a newborn, to commute between Rhode Island and New Hampshire, and collect the data without his help. Thank you for your optimism, your interest in my research and your support.

I am also greatly indebted to Professor Dmitri Nikshych for providing me with great quality instructions and teaching me Abstract Algebra. His teaching

and interest in Abstract Algebra structures allowed me to take the responsibility and study how other people understand Abstract Algebra concepts.

I would like to thank Professor Maria Basterra for increasing my interest in mathematics and mathematics education, for “being-a new-mom-and-researcher” example, and agreeing to be a member of my dissertation committee.

Again, I would like to thank all my dissertation committee members for their help and support.

I cannot overstate the role that Dr. Melissa Mitcheltree played in my life during my graduate study. I thank her for helping me to assimilate in a new learning environment, for teaching me how to be a graduate student, for supporting me and helping me with my study. I am very lucky to have a friend like her.

I would like to thank the Graduate School of University of the New Hampshire for giving me this wonderful opportunity. Special thanks to the Department of Mathematics and Statistics for providing me with teaching assistantship that gave me a unique teaching experience and financial support. This study was in part supported by the Graduate School Summer Fellowship. I thank all professors and graduate students of the department, and also all the students who agreed to participate in my study.

I am very grateful to my family. Without them I would have never start my graduate study and would have never completed this dissertation. Thank goes to my husband Andrei Smuk for his support, help, and inspiration for learning he gave me. My daughter Dasha who, being born before the dissertation was

complete, gave me a chance to graduate. Special thanks to my parents, Ludmila Titova and Sergey Titov, and grandparents, Nina Nenasheva and Ivan Nenashev for their life-long commitment to my personal and scholarly development.

Also I would like to thank all my friends for believing in me, for their personal support and help.

TABLE OF CONTENTS S IG N A TU R E P A G E ...ii D E D IC A T IO N ... iii A C K N O W L E D G E M E N T S ... iv LIST OF T A B L E S ... x LIST O F F IG U R E S ... xi A B S T R A C T ... xiv C H A PTE R PAGE I. IN T R O D U C T IO N ... 1 II. LITERATUR E R E V IE W ...4 Philosophical V iew ... 4

Classical View by J. P iaget... 5

Theories Based on Piagetian Idea of Reflective Abstraction... 10

Alternative views on the notion of abstraction... 14

Mathematical Knowledge Acquisition: Learning Abstract Algebra C oncepts 24 APO S Theory and Researches Based on It... 25

Studies framed in different perspectives... 31

III. TH E O R E TIC A L FR A M EW O R K ... 36

IV. R ESEA RCH Q U E S T IO N S ... 41

V. M E T H O D O L O G Y ...43

Introduction... 43

Settings and Instructional C o n text... 44

Data Collection Procedure 45 Students Artifacts...45 Students Interviews... 45 Observations... 47 Instruments... 47 Data Analysis...47 VI. DATA A N A L Y S IS ... 49 Quiz 1 ...49

Set - operation relations: universal quantifications... 51

Set - operation relations: closure... 56

Interesting response... 62

Quiz 2 ... 63

Set - operation relations. Groups and their subgroups... 63

Exam 1 ... 72

Identity - quantifiers... 73

Identity - uniqueness...78

Set - operation relations. Closure... 85

Concrete exa m p les ... 87

Identity element - group axioms...90

Exam 2 ... 93

Analysis of Problems 1 and 2 ... 94

Analysis of Problem 3 ...106

Analysis of Problem 4 ...115

Analysis of Problem 5 ...124

Interview II 140

Interview I I I ... 157

VII. D IS C U S S IO N S AND C O N C L U S IO N S ...165

S um m ary...166

Discussion... 167

Understanding concept of a binary structure...168

Making conclusions based on concrete objects: more cases of empirical generalization... 180

Definitions of objects. How students use th e m ... 184

Quantifiers... 187

Conclusions... 192

V III. IM PLIC A TIO N S OF THE S T U D Y ... 196

R E F E R E N C E S ... 199

A P P E N D IC E S ... 203

A P P E N D IX A DATA CO LLECTIO N TIM E T A B L E ... 204

A P P E N D IX B SAMPLES O F IN TE R V IEW Q U ESTIO N S AND W R ITTE N A S S IG N M E N T S ...205

A P P E N D IX C MAIN D EFIN ITIO N S AND TH E O R E M S U S E D ... 212

LIST OF TABLES

LIST OF FIGURES

Figure 1. The Process of Abstraction... 39

Figure 2. Student's definition of an associative operation...51

Figure 3. Student’s reasoning with the quantifier... 53

Figure 4. Student’s definition and quantifiers... 54

Figure 5. Student’s definition of operation on Z ...56

Figure 6. Student’s reasoning about division as a binary operation on Z ... 57

Figure 7. Student’s symbolical reasoning about division being binary operation on Z ....5 8 Figure 8. Choosing an operation... 59

Figure 9. Student’s definition of identity and reasoning about binary operations... 60

Figure 10. Student’s definition of binary operation with special property on Z ... 62

Figure 11. Student’s definition of a subgroup... 64

Figure 12. Missing “operational” part in subgroup definition... 65

Figure 13. Student’s definition of the General Linear Group GL(n, Q)...66

Figure 14. Student’s definition of the General Linear Group GL (n, Q )... 67

Figure 15. Student’s reasoning about subgroups. Confusion with operation... 68

Figure 16. Example of the change of the operation...70

Figure 17. Student’s definition of a subgroup. Missing “group” part...71

Figure 18. Missing “group” part example...72

Figure 19. Illustration of V 3 - 3 V problem in identity definition... 74

Figure 20. Correct order of quantifiers... 75

Figure 21. Response with misplaced quantifier...76

Figure 23. Missing quantifier... 77

Figure 24. Student’s definition of an identity element, stressing its uniqueness...78

Figure 25. Student’s solution for identity... 80

Figure 26. Student’s definition of an identity, stressing uniqueness...82

Figure 27. Multiple identity solution... 82

Figure 28. Student’s definition identity and its application... 84

Figure 29. Independent Identity element...85

Figure 30. Consideration of the closure...86

Figure 31. Student’s reasoning with concrete examples...88

Figure 32. Misleading "concrete” argument...89

Figure 33. Definition - group axiom controversy...91

Figure 34. Student’s reasoning about binary operation starting with group axioms 92 Figure 35. Student’s description of a cyclic group... 95

Figure 36. Student’s definition of a cyclic group... 95

Figure 37. Student’s definition lacking quantifier...96

Figure 38. Defining cyclic group as improper cyclic subgroup...97

Figure 39. Student’s reasoning about subgroups of Z i2... 98

Figure 40. Student’s definition of a cyclic group... 99

Figure 41. Definition of a cyclic subgroup without symbolic description of its elements. 100 Figure 42. Computations of subgroups of Z 12... 102

Figure 43. Computations of subgroups of Z 12 with references to some theorems 103 Figure 44. Reasoning about subgroups of Z 12, using the theorems...104

Figure 45. Reasoning with theorems... 105

Figure 47. Using sets which are not subgroups... 108

Figure 48. Using sets that are not closed under addition...109

Figure 49. Student’s Reasoning about subgroups...110

Figure 50. Student's argument with concrete examples... 111

Figure 51. Student’s attempt to prove her/his statement... 112

Figure 52. Supporting of the argument... 113

Figure 53. Finding counterexample... 114

Figure 54. Argument about finite and cyclic groups... 116

Figure 55. Reasoning about finite group and its subgroup...119

Figure 56. Table argumentation... 120

Figure 57. Proof of the concrete case...121

Figure 58. Proof by way of contradiction... 123

Figure 59. Misleading subgroup argument... 125

Figure 60. Proof of the statement about H being subgroup of G ... 126

Figure 61. Proof based on H being a subgroup...127

Figure 62. Student’s computations... 150

Figure 63. Student's proof...164

ABSTRACT

UNDERSTANDING OF ABSTRACT ALGEBRA CONCEPTS

by Anna Titova

University of New Hampshire, May, 2007

This study discusses various theoretical perspectives on abstract concept formation. Students’ reasoning about abstract objects is described based on theoretical proposition that abstraction is a shift from abstract to concrete. Existing literature suggested a theoretical framework for the study. The framework describes process of abstraction through its elements: assembling, theoretical generalization into abstract entity, and articulation. The elements of the theoretical framework are identified from students’ interpretations of and manipulations with elementary abstract algebra concepts including the concepts of binary operation, identity and inverse element, group, subgroup, cyclic group. To accomplish this, students participating in the abstract algebra class were observed during one semester. Analysis of interviews conducted with seven students and written artifacts collected from seventeen participants revealed different aspects of students’ reasoning about abstract objects. Discussion of the

analysis allowed formulating characteristics of processes of abstraction and generalization.

The data showed that the students often find it difficult to reason about abstract algebra concepts. They prefer to deal with “concrete” objects and often are confused if the problem is stated in more general terms. Moreover, number of students based their arguments about a certain object on their understanding of a concrete structure. For example, some students said that if integer 1 does not belong to a given structure then this structure cannot be a group. Also, since abstract algebra concepts are complex structures, participating students repeatedly missed some elements of these structures during problem solving. One of the frequently missed elements was quantification of objects. Students often were confused how to use quantifiers.

The study elaborates on these problems and offers theoretical explanations of the difficulties. The explanations, therefore, provide implications for instructions and future research.

INTRODUCTION

Abstract thought is considered to be the highest accomplishment of the human’s intellect as well as its most powerful tool (Ohlsson, Lehitinen, 1997). Most people consider mathematics as abstract and it is difficult to argue with this opinion. Even though mathematical problems can be solved by guessing, trial and error, experimenting (Halmos, 1982), there is still a need for abstract thought.

One can notice that there exists a fear of mathematics, when students believe that mathematics is not for them because they failed to understand something on the first try. There is support (Ferguson, 1986) to the hypothesis that abstraction anxiety is an important factor of mathematics anxiety, especially in topics which are introduced in the middle grades. Students’ statements such as “I understand 2 and 3, but I don’t understand x and y.” were observed by Ferguson (1986). I think if we understand the nature of abstraction, its acquisition, we can help students not to reduce the level of abstraction but, to the contrary, to bridge the gap from the abstract to concrete. This study aims to explore the process of abstraction, to give a description of its components and outcomes. My goal is to understand certain cognitive processes through empirical observations including classroom observations, interviews, and written artifacts collection. During the study I observed certain learning phenomena and made a theoretical analysis of these phenomena. “A theory should help us say

that if certain phenomena are observed, then other phenomena, are likely to occur as consequences” (Dubinsky, 2000a; p. 11). I select a qualitative approach to my research. I attempt to analyze students’ construction of knowledge (knowledge of abstract/mathematical object, in particular) in the tradition of a grounded theory (Charmaz, 2003; Glaser & Strauss, 1967) in the content of group theory.

The importance of knowing abstract algebra and group theory in particular, is widely acknowledged. Undergraduate students use mathematical ideas, learned in these courses, in many scientific areas, such as physics, computer science and chemistry. Abstract algebra is also is an essential part of middle and secondary school teachers’ preparation. However, students often find the course difficult and many researches indicate problems and gaps in students’ understanding of group theory concepts (Dubinsky et al, 1994, 1997; Hazzan, 1999; Nardi, 2000). Only recently has serious research been directed towards learning and teaching of abstract algebra: “A literature search revealed 15 articles on the learning of abstract algebra. Eleven of them had been published since 1994, of which 9 grew from the work of Dubinsky, Leron, and their collaborators” (Findell, 2001; p. 6). These arguments reason my choice of investigating students’ cognitive processes of abstraction and generalization in the context of group theory. The goal of this study is not to fill all possible gaps but to suggest some theoretical constructs which may help to understand and possibly avoid some of students’ difficulties in the future.

The following chapter reviews theoretical perspectives on the notion of abstraction and generalization and on students’ learning of group theory concepts. Chapter 2 describes theoretical guiding of the study or the theoretical framework. Chapter 3 and 4 propose research questions and methodology for searching answers. Detailed data analysis, illustrated by the data excerpts is described in Chapter 5, followed by the discussion of the findings and main conclusions in Chapter 6. Finally, I share my ideas for implications of the study for teaching and future research in Chapter 7.

CHAPTER I

LITERATURE REVIEW

This chapter includes a review of recent and classical theories on learning abstract objects. Studies that were guided by these perspectives are also discussed.

Philosophical View

There exist many definitions of abstraction. I would like to start this literature review with a brief description of the first known perspectives on this notion. Attempts by philosophers and cognitive psychologists to explore the meaning of the process of abstraction date back to Aristotle and Plato. If we think about an object as concrete as an apple it does not mean we necessarily must see the object in front of us. In other words, we would call this mental “apple” abstract. Abstract entity, in a philosophical sense, is “an object lacking spatiotemporal properties, but supposed to have being, to exist... Abstracta, sometimes collected under the category of universals, include mathematical objects, such as numbers, sets, and geometrical figures, propositions, properties, and relations... The abstract triangle has only properties common to all triangles” (Audi, 1999, p.3). From the empirical point of view, abstract ideas are universals, i.e. relations, types or properties of objects:

We should only know what is now present to our senses: we could not know anything about the past - not even that there was a past - nor could we know any truths about our sense-data, for all knowledge of truths, as we shall show, demands acquaintance with things which are of an essentially different character from sense-data, the things which are sometimes called 'abstract ideas', but which we shall call 'universals'. We have therefore to consider acquaintance with other things besides sense-data if we are to obtain any tolerably adequate analysis of our knowledge. (Russell, 1998, p.22)

For example (Russell, 1998) in geometry, when we wish to prove something about all triangles, we draw a particular triangle yet we do not to use any characteristics which it does not share with every other triangle. According to Russell,

The beginner, in order to avoid error, often finds it useful to draw several triangles, as unlike each other as possible, in order to make sure that his reasoning is equally applicable to all of them. But a difficulty emerges as soon as we ask ourselves how we know that a thing is a triangle. If we wish to avoid the universal triangularity, we shall choose some particular triangle, and say that anything is a triangle if it has the right sort of resemblance to our chosen particular. But then the resemblance required will have to be a universal (p.46).

Hence, for this philosopher, abstractions are created by extracting all properties common for all objects (universals), which the individual has experienced. These abstractions are needed to recognize a specific object among other objects.

Classical View by J. Piaget

Respecting the classico-philosophical point of view, Piaget (1970a, 1970b)

says that mathematical abstraction (as also abstraction, marked by

extraordinarily detailed and vivid recall of visual images, which gives rise to the knowledge of universals) implies certain operations, which were not considered by Aristotle.

Piaget (1970a) is searching for the answer to the very difficult and important question about the formation of human knowledge. Referring to the classical view of the problem researchers wonder if all cognitive information has its source in objects, so that the learner is “instructed” by the objects or another individual in the world outside him; or, on the contrary, the subject possesses a form produced or growing from within structures which the subject imposes on objects. The first assumption is coming from the traditional empiricism; the second is maintained by the varieties of a priories or innatism. Although these are two different statements, Piaget noticed common trends of established epistemologies: there exists a subject aware of its power in various degrees (even just a perception of objects); objects exist for a subject (even such object as ‘phenomena’); and the most important is a mediation between the subject and the object and vice versa.

Further, Piaget (1970a) studies characteristics of cognition: association and assimilation. He criticizes the concept of association by claiming that this concept only refers to an external bond between the associated elements (it does not give a learner a deep knowledge about object and its properties), “whereas the idea of assimilation implies that of the integration of the given within a prior structure or even a formation of a new structure under the elementary form of a scheme” (p.22).

Piaget distinguishes three aspects of the process of assimilation: repetition, recognition and generalization, which can closely follow each other. By repetition he understands a reproduction of the same movement (or reproductive

assimilation) and the formation of the beginning of a cognitive scheme; recognition or recognitive assimilation is the applying of previously created scheme to a new object or situation; and when the subject repeats the action in this new situation we deal with generalizing assimilation or generalization.

Even in the very beginning of an individual’s cognitive development one can observe the construction of new combinations by a union of abstractions derived either from objects themselves, or from schemes of actions applied to them. For Piaget (1970a) the child’s recognition of an object, as something having specific properties, requires an abstraction starting from objects. On the other hand learners’ coordination of means and ends, taking to account the proper sequence of required movements, is a new form of behavior “compared with the global acts...” (p.23); but this new behavior is naturally acquired from such acts by a process of deriving from them the relations of order, overlapping, etc., necessary for this coordination. This coordination Piaget does no longer consider as appearance of abstraction from concrete object but rather as an abstraction, which derives higher-order structures from the previously acquired lower-order structures.

Working further, Piaget (1970b) studies the details of the notion of abstraction and discusses it in terms of mathematical knowledge formation and logic. The author agrees that logical and mathematical structures are abstract, whereas physical knowledge - the knowledge based on experience - is concrete. He is struggling to find answers to the question about human knowledge formation and, since mathematical and logical structures are defined

as abstract structures, he asks: “What are these structures abstracted from?” One view on abstraction comes from an empirical ideology, that is - our knowledge is derived from the concrete object itself; in this case the question remains - What are the concrete objects in mathematics? A second view claims that, since the transformation of the object can be carried mentally, we can take into the account the actions itself or our actions upon the object. In that way the abstraction is derived from the action itself. This position seems for Piaget as a basis of logical and mathematical abstraction. The first type of abstraction from object he defined as a simple abstraction, for instance, when a child lifts objects and realizes that smaller objects are usually lighter then bigger objects, so the idea of weight, as a characteristic of an object is abstracted from the objects themselves. The second type he called the reflective abstraction, for example when a child counts five marbles in different ways, he notices that it does not matter how he places them, he always gets the same number. This way the child discovers the mathematical property of addition - commutativity, and this knowledge is drawn not from the physical properties of marbles but from the actions the child carried out on the marbles. Piaget uses the word “reflective” in a double sense in terms of psychology and, in addition, of physics. The following citation helps to understand how Piaget makes a distinction between the two types of abstraction,

On the one hand, there are individual actions such as throwing, pushing, touching, rubbing. It is these Individual actions that give rise most of the time to abstraction from object. This is the simple type of abstraction...Reflective abstraction, however, is based not on individual actions but on coordinated actions. Actions can be coordinated in a number of different ways. (Piaget, 1970, p. 18)

The intermediate step between empirical and reflective abstraction that occurs after the action(s) have taken place on the object, Piaget calls a pseudo- empirical abstraction. During pseudo-empirical abstraction, the subject engages with an external object and extracts properties of the actions introduced into the object during empirical abstraction.

From this argument it is necessary to suppose that abstraction starting from actions and operations - reflective abstraction - differs from abstraction from perceived objects - simple abstraction (in the book “Mathematical Epistemology and Psychology” (Beth, Piaget, 1966) Piaget calls simple abstraction - “empirical abstraction”, p. 188-189) - in the sense that reflective abstraction is constructive, while on the contrary simple or empirical abstraction consists of deriving commonalities from class of objects by combination of abstraction and simple generalization. By generalization Piaget (1966, p.243) means “the simple observation that several objects posses a common character.” Thus, Piagetian position on the process by which the subject derives new knowledge from the results of his/her own actions is as follow:

a) logico-mathematical experience consists of observing actions performed upon any object;

b) the results are determined by the schemes of the actions;

c) in order to observe these results, the subject has to perform other actions using the same schemes as those the product of which must be examined;

e) the abstraction, by means of which the subject acquires new knowledge as a result of his actions - involves some construction. (Beth, Piaget, 1966)

From Piagetian (1966, 1970a, 1970b) view, the axiomatization is based on the reflective abstraction. It occurs when a thinker derives conceptually elementary principles, for example identities. In the early stages the axioms were still accepted as intuitive and were borrowed from the natural thought, but later theories become less and less intuitive. More precisely,

If we analyze the reflective abstraction into ‘reflection’ in the quasy-geometric sense of the projection of certain previously given relationships on to a new plane of thought, and ‘reflection’ in the noetic (originating in the intellect) sense of reorganization necessitated by the reconstruction of these relationships on this new plane, then this later aspect prevails over the former (p. 64).

Therefore, according to Piaget, even before the first mathematical entity was formed, the process of reflective abstraction gives rise to the initial concepts and operations in mathematics.

Theories Based on Piagetian Idea of Reflective Abstraction

In his papers about advanced mathematical thinking, Dubinsky (1991a, 1991) proposes that the concept of reflective abstraction, introduced by Piaget, can be a powerful tool in the process of investigating mathematical thinking and advanced thinking in particular.

Dubinsky (1991a) presents a brief description of his theory of mathematical knowledge and its acquisition in the area of mathematics that is more advanced then Piaget considered for his research. He takes a view that knowledge and its acquisition are not easily distinguishable. According to

Dubinsky there are three aspects which must be investigated in order to understand mathematical knowledge and its acquisition: problem situations, schemas (more or less coherent collections of cognitive objects and internal mental processes for manipulating these objects), and responses.

One must consider the difference in the problem situation as it is intended by the observer and as it appears to the subject. One must understand the nature of schemas and the means by which they are constructed. Finally, it is necessary to explain how the subjects select the schema to be used in the response and what determines the kinds of new constructions (if any) that are made (p.5).

Elaborating further, Dubinsky (1991a) uses two observations made by Piaget to form his general theory: 1) reflective abstraction is present in the very early ages in the coordination of sensory-motor structures; 2) the entire history of the development of mathematics may be considered as an example of the process of reflective abstraction. Although Piaget concentrated on the development of mathematical knowledge at the early age, Dubinsky suggests that the same approach can be extended to more advanced undergraduate mathematical topics such as mathematical induction, predicate calculus, functions, topological spaces and vector spaces, etc., so they all can be analyzed in terms of extensions of the same notions Piaget used. Dubinsky lists various kinds of construction in reflective abstraction, which is heavily based on the work by Piaget: interiorization - the process of construction internal processes as a way of making sense out of perceived phenomena; coordination of two or more processes to construct a new one; encapsulation of a dynamic process into a static object; generalization of existing construction to a wider collection of phenomena; reversing the original processes to construct a new process. The

final construction process of reflective abstraction is proposed by Dubinsky (not Piaget).

Dubinsky (1991) extends Piagetian ideas and reorganizes them into a general theory of mathematical knowledge and relates his theory in specific mathematical topics, such as vector spaces and functions, to explain some processes which may occur in the learning process. He proposes a notion of genetic decomposition. According to Dubinsky (1991, p. 96), the genetic decomposition of a concept is a description of the mathematics involved and how a subject might make the construction(s) that would lead to an understanding of the concept. Dubinsky describes examples of genetic decomposition for mathematical induction, predicate calculus, and function.

In terms of educational implications, Dubinsky’s position is that learning consists of applying reflective abstraction to existing schemas in order to construct new schemas for understanding concepts, thus the schema can not be constructed in the absence of previously existed schemas. In this way Dubinsky suggests the following instructional approach to foster conceptual thinking in mathematics: 1) observe students to see their developing of concept images; 2) analyze the data and develop a genetic decomposition for each topic; 3) design instruction that moves students through the steps of the genetic decomposition, develop activities that will induce students to make a specific reflective abstractions; 4) repeat the process and continue until stabilization occurs.

Another line of research based on Piagetian theory of reflective abstraction is a study conducted and developed by Goodson-Espy (1998). She

observed abstraction process during problem solving and examined the transition that students make from arithmetic to algebra. Although her study is framed in Sfard’s (1991) theory of reification, Goodson-Espy’s framework includes an idea of reflective abstraction proposed by Piaget and, further, the notion of levels of reflective abstraction, proposed by Cifarelli (as cited in Goodson-Espy, 1998). The first level is defined as Recognition - the ability to recognize characteristics of a previously solved problem in a new situation and to believe that one can do again what one did before. The second level of reflective abstraction is Re­ presentation. At this level the student becomes able to run through a problem mentally and is able to anticipate potential sources of difficulty and promise. The third level of reflective abstraction is Structural abstraction. Structural abstraction occurs when a student evaluates solution prospects based on mental ‘run- throughs’ of potential methods as well as methods that have been used previously. Goodson-Espy in her paper suggests that there are strong relations between the theory of reification and the levels of reflective abstraction. Moreover, she suggests that Cifarelli’s levels of reflective abstraction may be used to illustrate how the transition from one stage of concept formation (from reification theory) to the next stage could take place.

Theoretical framework of Nardy’s (2000) study lies primarily in the Piagetian concept of reflective abstraction. The author explores the difficulties in students understanding of abstract algebra, group theory, in particular. She refers this understanding to advanced mathematical thinking in Dubinsky’s (1991, 1991a) sense. The study reported conceptual difficulties with some concepts of

group theory, which resonates with the findings in the area of investigation of learning advanced mathematics. The author and her colleagues are involved in further research in this area.

In his unpublished work, Harel (1995) elaborates more on the notion of abstraction and exemplifies Piagetian empirical (or simple) and reflective abstractions using epistemology of the concept of function. In this paper Harel illustrates some aspects of the theory using examples of interviews he conducted with college students and his observations.

Alternative views on the notion of abstraction

Recently several authors have critically analyzed classical and Piagetian approach and proposed alternative outlook on the notion of abstraction. (Hershkowitz, Schwarz, Dreyfus, 2001; Ohlsson, Lehitinen, 1997; Mitchelmore and White, 1995, 1999; Harel and Tall, 1991).

Making a fresh start, Ohlsson and Lehitinen (1997) approached the problematic of high-order cognition via distinguishing abstraction and generalization. This observation led Ohlsson and Lehitinen (1997) to the reevaluation of the role of generality in learning process. For them “to generalize” means to extract “commonalities from exemplars” (p. 38), while the main cognitive function of abstraction is to enable the assembly of previously existed ideas into more complex structure. Ohlsson and Lehitinen (1997) suggest that “people experience particulars as similar precisely to the extent that, and because, those particulars are recognized as instances of the same abstraction" (p. 41).

By reviewing the classical Aristotelian ideas Ohlsson and Lehitinen (1997) bring our attention to the fact that scientific theories (as good examples of higher order knowledge) do not fit the generalization idea:

Consider for example, the law of mechanical motion. On the Aristotelian view, Isaac Newton shouid have arrived at the equation

F = m x a

by measuring F, m and a many times in different situations and noticing that the product of m and a equals F in each instance, (p. 38)

The formulation of Darwin’s theory, for example, preceded its application to particular cases; hence, this theory can not be generalization in the classical sense. Ohlsson and Lehitinen (1997) conclude:

In summary, important examples of higher order knowledge in science, mathematics, and other fields are not, in fact, created by extracting commonalities across particular objects or events. In case after case, key ideas...were not as a matter of historical fact, discovered via generalization and could not, even in principle, be discovered that way (p. 40).

Ohlsson and Lehitinen (1997) claim that in order to recognize an object as an instance of an abstraction, the learner must already possess that abstraction. In other words, adopting Hayek’s (1952, 1978) terminology, the abstract has

primacy over the concrete. According to Hayek (1952, pp. 42-43; 1978, pp. 165-

172), the general concept is a “presupposition” of experience rather than the product of abstraction from what is presented in experience.

The authors assume that the deep idea is complex, i.e. has other ideas as parts. For example, to understand the idea of a group the learner must have an idea of a set, function, and a binary operation. The observation about complex idea suggests that new ideas are created by assembling previously acquired ides into new structure. As a result of assembling we have a new structure, more

abstract so is their combination. Hence, according to Ohlsson and Lehitinen, we create a new abstraction, operating on existed abstractions, not on concrete experience.

In this case assembling is not mere association - a link between cue and associate, where activation of the cue evokes the association (Halford at at., 1997). Halford distinguishes two types of associations: (1) Elemental association, which comprises links between pairs of entities; and (2) Configural association, which entails two stimuli each of which modifies the link between the other stimulus and the response. The first type does “not require any representation other than input and output and, therefore, cannot achieve any abstraction” (p.22); the second type “cannot support transfer between problem isomorphs. Therefore, a configural association can achieve only the minimal level of abstraction” (p.22); on the contrary, the result of assembling is a more complex idea - an abstract idea. The application of this complex abstract idea moves from the abstract toward the concrete. The term articulation is used by Ohlsson and Lehitinen (1997) to refer to this process. In other words, articulation is a process through which “abstract schema (knowledge structures, which regulate thinking which goes beyond immediate experience) functions as a plan, a form to be filled with content” (Ohlsson, 1993, p. 61).

In the present view, abstraction is prerequisite for learning, whereas generalization is a product of learning process. In fact, abstract ideas are generated from other abstract ideas. But how do learners acquire these

abstractions in the first place? Ohlsson and Lehitinen (1997) review several possible answers:

(1) Postulating innate abstraction.

The universal structure of a single object becomes a figure. For example, “mathematicians know only one number system, but careful analysis has produced a representation that has become an object of inquiry in its own right. The result is a new field, abstract algebra” (p.44).

(2) Seeking the origin of initial abstraction in discourse. (3) Process of induction.

In summary, two cognitive processes were postulated by Ohlsson and Lehitinen (1997). First, to learn a complex idea is to assemble available abstract ideas into new structure. This process moves from the simple toward complex (not from concrete to general). Second, abstractions are applied to concrete objects via articulation. This process moves from abstract toward concrete.

A soviet educator Davydov (1972/1990) approaches the problem of human cognition by distinguishing empirical and theoretical thoughts. To clarify the meaning and the difference between the two kinds, the author starts with the analysis of the nature of human thought in traditional formal logic. In this view, subject is extracting similarities from the set of particular objects (which exists independently of subject), so that particular objects can be combined into a class after comparison according to the sort of similar properties. A class is a mental formation - repeating properties of many objects, which has become a particular and independent object of thoughts. Thus, formal logic defines (formal)

abstraction and (formal) generalization as processes of identifying sensorially given, observable, external properties of an individual object.

Formal logic considers a thought as a transition from concrete and individual to the abstract together with the reverse transition. Concrete (as distinct from abstract) is defined as individually given, directly observable object itself. The thought that accomplishes previously mentioned transitions through formal generalization and abstraction forms empirical concepts. This theory is usually called the empirical theory of thinking (the notion is proposed by an English philosopher John Locke) and in these terms formal abstraction and generalization are called empirical abstraction and generalization. These processes solve problems of classifying objects by the external attributes and problems of identifying these attributes.

Davydov (1972/1990) is concerned with limitations of the empirical interpretation of abstraction and generalization. It follows from the definition, that in science, for example, traditional empirical abstraction and generalization is limited by directly observable phenomena. In general, the fundamental weakness of empirical theory is that every concept can be reduced to some concrete data. It means that we can find the appropriate concrete for any abstract attribute. From this position students can learn only what they can observe and experience (together with the teacher’s knowledge, which is imposed on students’ life experience).

Davydov (1972/1990) argues that scientific knowledge is not a simple extension and expansion of people’s everyday experiences. “It requires the

cultivation of particular means of abstraction, a particular analysis, and generalization, which permits the internal connections of things, their essence and particular ways of idealizing the objects of cognition to be established” (p. 86). Following this argument, the author proposes a theoretical approach to the theory of thinking - theoretical abstraction and generalization.

Theoretical abstraction is a theoretical analysis of objects (concrete or previously abstracted) and construction of a system that outlines the whole picture of the new concept being studied so it is ready to be applied for the correct recognition of particular objects. Theoretical generalization defined as a process of identification of deep, structural similarities, which identify the inner connections with previously learned ideas. According to Davydov (1972/1990), theoretical abstraction is linked to theoretical generalization in a following way: theoretical abstraction starts from initial abstracts - ready-made empirical abstractions; “the investigator can find it only in studying actual data and their relationships” (p.289). Further, from the simple, undeveloped, inconsistent first form of abstraction, the development proceeds with the analysis of these initials to obtain the necessary (theoretical) generalizations, which then will be synthesized to obtain a consistent final form - abstract idea. So, for Davydov, the process of abstraction does not proceed from concrete to abstract, but from undeveloped to developed abstraction, which allows learners to see the new features in concrete objects, when this abstraction is applied.

If the transition from general and abstract to particular has been mastered, then students bridge the gap between the concrete and the abstract. For

Davydov (1972/1990), “the more abstract the initial generalization, the more concretization its thorough mastery requires” (p.23). In other words, the stage of transition from abstract to particular should include more concrete problems for better articulation. The development of abstract, thus, depends on the accumulation of conceptions and perceptions.

In summary, Davydov (1972/1990) distinguishes two types of abstraction and generalization: theoretical and empirical. However, theoretical and empirical processes are linked to each other. From his argument it follows that for the learning of mathematics empirical theory is not enough. The main characteristics of empirical and theoretical thoughts are summarized in Table 1:

Empirical Simple External

Theoretical Complex Internal

Table 1. Characteristics of Thoughts.

Mitchelmore and White (1999) constructed the theoretical framework following Davydov’s principles of generalization and abstraction, borrowing the notion of content-related or theoretical generalization. Also Hershkowitz, Schwarz, Dreyfus (2001) proposing an approach to the theoretical and empirical identification of a process of abstraction, build their functional definition of abstraction on Davydov’s theory.

Mitchelmore and White (1995) noted that students often divide mathematical problems into two categories: “abstract” and “real-life” or “concrete” problems. Although “concrete” usually means an easier problem - a problem involving concrete objects, students seemed to prefer “abstract” exercises, given

in a symbolic form. On the other hand, according to the authors, there is strong evidence that abstractness of mathematics is its well-known difficulty. Ideas become more difficult as they become more abstract. Mitchelmore and White see a conflict here and to find an explanation for this phenomenon they turn to the very definitions of abstraction and abstract:

Abstract (adj): Apart from the concrete; general as opposed to particular;

expressed without references to particular examples.

Abstract idea: Mental representation or concept that isolates and generalizes an

aspect of an object or group of objects from which relationships may be perceived.

Abstract (ver.): To consider apart from particular instances; to form a general

notion of.

In terms of these definitions, the authors distinguish two types of abstraction: “abstract-general” - when a mathematical idea is linked to concrete objects (or other mathematical ideas); and “abstract-apart” - when a mathematical idea is separated from the context.

Reviewing the theories and studies of abstraction conducted by other researchers, the authors derived a general abstraction cycle: recognition —»•

manipulation —► reification, where by reification they mean a process of

converting a concept into an object of thought, extending Sfard’s definition. Mitchelmore and White believe that the degree of abstraction increases as their “abstraction cycle” is repeated several times:

For example, the physical act of counting is reified to whole numbers. Other numbers such as fraction and negative numbers are then reified and a general concept of number emerges. Study of the properties of number systems and other similar structures leads eventually to concepts such as groups, fields and spaces, each concept encapsulating a specific set of properties present in the various systems. Finally, the concept of a category is formed to abstract the common features of all such structures. Each step is an abstraction, and each new concept is experienced as more abstract than the concepts from which it is abstracted.

Dienes (as cited in Mitchelmore and White, 1995), states that the degree of abstraction of a concept is in direct proportion to the amount of variety of the experiences from which it has been abstracted. Hiebert and Lefevre (as cited in Mitchelmore and White, 1995) state that abstractness increases as knowledge becomes freed from specific contexts. In Mitchelmore’s and White’s opinion those statements are equating abstractness and generality.

Mitchelmore (1994), stimulated by Skemp’s (as cited in Mitchelmore 1994) work, proposed a model of conceptual development, consisting of two important phases: abstraction and generalization. Generalization appears as a never- ending process as more and more situations are brought in under the same abstraction. Later, however, White and Mitchelmore (1999) critically analyzed generalization as a shift from concrete to abstract, where students are involved into their pattern-seeking activities. They pointed out that, from this perspective, teaching model states “always proceed from particular to general”. This model is criticized by White and Mitchelmore with support of Davydov’s definition of generalization. The critique comes from the statement that classification of objects on the basis of external characteristics does not identify inner connections.

In summary, the authors believe in “abstract - to - concrete” (“general - to - particular”) learning order. Moreover, they do not equate abstraction and generalization

We note that the terms generalization and abstraction are often used interchangeably in the literature. The essential difference, as we see it, is that abstraction creates a new mental object (a concept) whereas generalization extends the meaning of an existing concept. The act of abstracting is based on generalizing, but is seen as qualitatively different from simply identifying patterns

in a set of examples. It is a many to one function where generalizations are synthesized from many inputs to form a new abstraction (p. 5).

Harel and Tall (1991) were also investigating the meaning of the terms abstraction and generalization. They defined generalization as a process of applying a given argument in a broader context. In mathematics, in particular, this process depends on the individual’s current knowledge. From their observations of the students, the authors distinguish three different kinds of generalization, depending on individual’s mental constructions:

1. Expansive generalization occurs when the subject expands the applicability range of an existing schema without reconstructing it.

2. Reconstructive generalization occurs when the subject reconstructs an existing schema in order to widen its applicability range.

3. Disjunctive generalization occurs when, on moving from the familiar context to a new one, the subject constructs a new, disjoint schema to deal with the new context and adds it to the array of schemas available, (p. 38)

The last type of generalization is not considered by Harel and Tall as a cognitive generalization “in the sense that the earlier examples are not seen by the individual as special cases of the general procedure” (p. 38), however the first two seem for them more appropriate for cognitive development. Also they argue that in a short term expansive generalization is cognitively easier, but in a long run there are times when reorganization of knowledge becomes essential which means that reconstructive generalization becomes more appropriate.

Harel and Tall (1991) define abstraction as a process which occurs when subject focuses attention on specific properties of a given object and considers these properties as isolated from the original. The authors attribute the abstraction theory to reconstructive generalization, “because the abstracted properties are reconstructions of the original properties, now applied to broader

domain” (p.39). In mathematics, abstraction of specific properties to form the basis of the definition of a new mathematical object is one constituent of the two distinct processes which form the process of formal definition. The second process is construction of an abstract concept through logical deduction from the definition. Harel and Tall call the first process formal abstraction - the abstraction of a new concept through the selection of properties of one or more specific situations. They, however, admit the difficulty of a formal abstraction for the learner and to help students to pass the difficulties they introduce another form of abstraction - generic abstraction. In this case concept formation starts with so- called prototypes - more specific examples, so students can see the properties required for the new concept and apply it to a wider range of examples, embodying an abstract concept.

The next section discusses how some of the theoretical perspectives can be applied to the abstract algebra content. It reviews studies of abstract knowledge acquisition focusing on abstract algebra courses.

Mathematical Knowledge Acquisition: Learning Abstract Algebra Concepts

The first undergraduate course of Abstract Algebra is always the great concern of mathematics department communities. It is explained by the importance of concepts and methods of problem solving in abstract algebra and the obvious learning difficulties which most of students usually experience. Clark et al. (1997) assumes that “perhaps even more troubling is the fact that during this course many of these students come to dislike mathematics even though, for a variety of reasons, mathematics continues to be their major. It seems this is

especially the case for many pre-service secondary mathematics teachers.” Indeed, the students develop a negative attitude toward mathematics in general and a fear of abstraction. With their mathematical background the students often have little experience thinking about the concepts that are dealing with structures, or proving theorems. Secondary education reforms or/and specially designed undergraduate courses are aimed to bridge this gap in students’ learning. Another approach to solving the problems with the abstract algebra course was started in the late 1980s by Ed Dubinsky and his colleagues (Clark at al., 1997). They chose to confront the problems they saw in the content and pedagogy of traditional abstract algebra courses by applying a framework for curriculum development and research in mathematics education that they had been developing for several years. This section introduces the main ideas of the framework.

APOS Theory and Researches Based on It

In recent years, mathematics education community started to work on developing a theoretical framework and a curriculum for undergraduate mathematics education. Asiala et al. (1996) reported the results on their work in this area. The authors are concerned with theoretical analyses which model mathematical understanding, instruction based on the results of these analyses, and empirical data, both quantitative and qualitative, that can be used to refine the theoretical perspective and assess the effects of the instruction. Finally, the authors of this article are in the process of producing a number of studies of

topics in calculus and abstract algebra (Zazkis & Dubinsky, 1996; Dubinsky at al, 1994; Brown at al, 1997; etc) using their framework.

For the author’s research framework, research begins with a theoretical analysis. This initial analysis is based primarily on the researchers' understanding of the concept in question and on their experiences as learners and teachers of the concept. Then the analysis informs the design of instruction. Implementing the instruction provides an opportunity for gathering data and for reconsidering the initial theoretical analysis with respect to this data. These repetitions are continued for as long as it appears to be necessary to achieve stability in the researchers' understanding of the epistemology of the concept.

The authors noted that each time the researcher cycles through the components of the framework, every component is reconsidered and, if possible, revised. In other words, the research builds on previous implementations of the framework.

Based on the theories of cognitive construction developed by Piaget for younger children, Dubinsky proposed APOS (action - process - object - schema) theory. In terms of this framework, understanding of a mathematical concept begins with manipulating previously constructed mental or physical objects to form actions; actions are then interiorized to form processes which are then encapsulated to form objects. Objects can be de-encapsulated back to the processes from which they were formed. Finally, actions, processes and objects can be organized in schemas. This assumption about mental constructions is based on a specific notions described by Piaget.

In a more detailed discussion, the authors explain that “an action is a transformation of objects which is perceived by the individual as being at least somewhat external”, (p.9). In the context of Abstract Algebra, for example, if the elements of a group can be listed explicitly, then it is not difficult to find its subgroup and work with cosets. “Understanding a coset as a set of calculations that are actually performed to obtain a definite set is an action conception.” (p. 10). However, more is required to work with cosets in a group such as Sn, the group of all permutations on n objects where simple formulas are not available. In terms of the theoretical framework being discussed, students who have no more than an action conception will have difficulty in reasoning about cosets: “In the context of our theoretical perspective, these difficulties are related to a student's inability to interiorize these actions to processes, or encapsulate the processes to objects.” (p. 10)

Further, when an action is repeated, and the individual reflects upon it, it may be interiorized into a process. In abstract algebra, a process understanding of cosets includes thinking about the formation of a set by operating a fixed element with every element in a particular subgroup.

When an individual reflects on operations applied to a particular process, becomes aware of the process as a totality, realizes that transformations, can act on it, and is able to actually construct such transformations, then he or she is thinking of this process as an object. In an abstract algebra context, given an element x and a subgroup H of a group G, “if an individual thinks generally of the (left) coset of x modulo H as a process of operating with x on each element of H,

then this process can be encapsulated to an object xH. Then, cosets are named, operations can be performed on them and various actions on cosets of H, such as counting their number, comparing their cardinality, and checking their intersections can make sense to the individual”, (p. 11).

A collection of processes and objects can be organized in a structured manner to form a schema. Schemas, at the same time, can be treated as objects and included in the organization of "higher level" schemas.

In order to illustrate that the discussed mental constructions take place during learning mathematical concepts, Asiala et al. (1996) suggest to gather data using three kinds of instruments: written questions and answers in the form of examinations in the course or specially designed question sets; in-depth interviews of students; and a combination of written instruments and interviews. Their written instruments contain fairly standard questions about the mathematical content and they are analyzed in relatively traditional ways. This information shows what the students may or may not learn. It also illustrates the possible mental constructions. To access the full range of understanding, the authors select interviewee’s group by including students who gave correct, partially correct, and incorrect answers on the written instruments. They also routinely select students who appear to be in the process of learning some particular idea rather than those who have clearly mastered it or those who had obviously missed the point.

The most important part of qualitative research is to collect the needed data and analyze it properly. Asiala, et al. (1996), in their framework, divided data

analysis into 5 steps: 1. Script the transcript. 2. Make the table of content. 3. List the issues. By an issue they mean some very specific mathematical point, an idea, a procedure, or a fact, for which the interviewee may or may not construct an understanding. For example, in the context of group theory one issue might be whether the student understands that a group is more than just a set, that is, it is a set together with a binary operation. 4. Relate to the theoretical perspective. At this step theoretical perspectives are revised. 5. Summarize performance. “The mathematical performance of the students as indicated in the transcripts is summarized and incorporated in the consideration of performance resulting from the other kinds of data that are gathered”(p. 27). It should be noted that in the design of instruction, the authors specially highlight the use of cooperative learning and computer programming language.

To highlight the importance of created framework, Dubinsky (2000) makes the point that working with abstraction in mathematics in general can help students to understand some complex situations with which they have to deal in everyday life, so that APOS theory can help to explain why people have difficulty in understanding some aspects of everyday life in society. The described theoretical framework is illustrating the theoretical and methodological approach which was used by the number of authors in their study of students’ knowledge acquisition in abstract algebra. According to Clark et al. (1997), there are two central questions which can be explored in light of these theoretical perspectives: (1) if students’ attitude toward mathematics in general and abstract algebra in particular has been improved as a result of a new treatment; and (2) what